3-D Numerical Mode Matching Method for Off- Centered Electromagnetic Well Logging Tools in Noncircular Vertical Borehole and Invasion Zones in Multilayered Media

In electromagnetic (EM) well logging for petroleum exploration, off-centered tools within a noncircular borehole and invasion zones in multilayered media represent a challenging 3-D problem for traditional numerical methods. The 3-D finite-element numerical mode-matching (FNMM) method and the spectral numerical mode-matching (SNMM) method are developed to address this problem in this work. The numerical mode-matching (NMM) methods reduce the original 3-D well logging problem into a series of 2-D open waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with anisotropic inhomogeneous media in the horizontal directions are solved numerically by the mixed finite-element method (MFEM) with the flexibility for the complex geometry and the mixed spectral element method (MSEM) with the exponential convergence. A general operator form of Maxwell’s equations is also derived to obtain the source excitation vector independent of the $z$ -axis. Therefore, the NMM solvers can effectively and accurately simulate the response of off-centered EM well logging tools in an irregular borehole in complex formations. The NMM methods are applied to simulate three formations including a five-layer isotropic homogeneous formation, an off-centered logging tool in a noncircular borehole with anisotropic invasion zones in a five-layer anisotropic formation, and the Oklahoma formation. Numerical experiments indicate that the NMM methods are highly efficient and accurate for the EM well logging models.